\[\frac{1}{x + 1} - \frac{1}{x}\]
Test:
NMSE problem 3.3.1
Bits:
128 bits
Bits error versus x
Time: 3.9 s
Input Error: 5.5
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\frac{\frac{-1}{x}}{1 + x}\)
  1. Started with
    \[\frac{1}{x + 1} - \frac{1}{x}\]
    5.5
  2. Using strategy rm
    5.5
  3. Applied frac-sub to get
    \[\color{red}{\frac{1}{x + 1} - \frac{1}{x}} \leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 1}{\left(x + 1\right) \cdot x}}\]
    4.6
  4. Applied simplify to get
    \[\frac{\color{red}{1 \cdot x - \left(x + 1\right) \cdot 1}}{\left(x + 1\right) \cdot x} \leadsto \frac{\color{blue}{x - \left(1 + x\right)}}{\left(x + 1\right) \cdot x}\]
    4.6
  5. Applied simplify to get
    \[\frac{x - \left(1 + x\right)}{\color{red}{\left(x + 1\right) \cdot x}} \leadsto \frac{x - \left(1 + x\right)}{\color{blue}{x + {x}^2}}\]
    4.6
  6. Using strategy rm
    4.6
  7. Applied square-mult to get
    \[\frac{x - \left(1 + x\right)}{x + \color{red}{{x}^2}} \leadsto \frac{x - \left(1 + x\right)}{x + \color{blue}{x \cdot x}}\]
    4.6
  8. Applied distribute-rgt1-in to get
    \[\frac{x - \left(1 + x\right)}{\color{red}{x + x \cdot x}} \leadsto \frac{x - \left(1 + x\right)}{\color{blue}{\left(x + 1\right) \cdot x}}\]
    4.6
  9. Applied *-un-lft-identity to get
    \[\frac{\color{red}{x - \left(1 + x\right)}}{\left(x + 1\right) \cdot x} \leadsto \frac{\color{blue}{1 \cdot \left(x - \left(1 + x\right)\right)}}{\left(x + 1\right) \cdot x}\]
    4.6
  10. Applied times-frac to get
    \[\color{red}{\frac{1 \cdot \left(x - \left(1 + x\right)\right)}{\left(x + 1\right) \cdot x}} \leadsto \color{blue}{\frac{1}{x + 1} \cdot \frac{x - \left(1 + x\right)}{x}}\]
    4.6
  11. Applied taylor to get
    \[\frac{1}{x + 1} \cdot \frac{x - \left(1 + x\right)}{x} \leadsto \frac{1}{x + 1} \cdot \frac{-1}{x}\]
    0.1
  12. Taylor expanded around 0 to get
    \[\frac{1}{x + 1} \cdot \frac{\color{red}{-1}}{x} \leadsto \frac{1}{x + 1} \cdot \frac{\color{blue}{-1}}{x}\]
    0.1
  13. Applied simplify to get
    \[\color{red}{\frac{1}{x + 1} \cdot \frac{-1}{x}} \leadsto \color{blue}{\frac{\frac{-1}{x}}{1 + x}}\]
    0.1

Original test:


(lambda ((x default))
  #:name "NMSE problem 3.3.1"
  (- (/ 1 (+ x 1)) (/ 1 x)))